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I think it's probably a seed pod. Many people think they are pretty and leave them on the plant, others take them off. Hopefully these articles will help: https://www.gardeningknowhow.com/ornamental/vines/clematis/growing-clematis.htm

http://www.britishclematis.org.uk/seedlist3.htm

From what you've described, it sounds like porcelain berry. This article has description, culture and pictures: http://www.missouribotanicalgarden.org/gardens-gardening/your-garden/plant-finder/plant-details/kc/r350/ampelopsis-brevipedunculata-var.-maximowiczii-elegans.aspx

These two links should answer your question: https://www.gardeningknowhow.com/ornamental/vines/clematis/propagate-clematis-cuttings.htm

http://www.clematisinternational.com/seeds.html

It really depends on what variety of clematis you have. This article will guide you on how to prune yours:
https://extension.missouri.edu/extensioninfonet/article.asp?id=1452

Hold off pruning until early spring for most types or following bloom in others. While it should not hurt the vine, depending on the type you have, blooming may be delayed or skipped entirely the following season if cut back too early. It may just be in need of some fertilizing too, which should help give the plant a jump start with growing.

This is really up to you, though planting one on either side will speed up the coverage. For additional information on growing and caring for clematis vines, these articles will help: https://www.gardeningknowhow.com/ornamental/vines/clematis